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I'm trying to reproduce the following solution to the ODE, $$\frac{d^{2}u}{d\rho^{2}} = \frac{l*(l+1)}{\rho^{2}}u$$

Solution: $$u\left(\rho\right) = C\rho^{l+1}+D\rho^{-l}$$

What I've tried in Mathematica using DSolve is

DSolve[u''[ρ] == (l*(l + 1))/ρ^2*u[ρ], u[ρ], ρ]

However, when I try using Simplify or FullSimplify on my Mathematica output, I can't get it into the simple version as above.

The Mathematica Output Solution that I get from the use of DSolve is

{{u[ρ] -> ρ^(1/2 I Sqrt[l] Sqrt[1 + l] (-(I/(Sqrt[l] Sqrt[1 + l])) - Sqrt[-4 - 1/(l (1 + l))]))C[1]
    + ρ^(1/2 I Sqrt[l] Sqrt[1 + l] (-(I/(Sqrt[l] Sqrt[1 + l])) + Sqrt[-4 - 1/(l (1 + l))]))C[2]}}

Thanks very much for your valuable time.

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You just have to add the assumption that $l,\rho\ge 0$:

Assuming[{l >= 0, ρ >= 0}, 
 Simplify[DSolve[u''[ρ] == (l*(l + 1))/ρ^2*u[ρ], 
   u[ρ], ρ]]]

(* ==> {{u[ρ] -> ρ^(1 + l) C[1] + ρ^-l C[2]}} *)
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